Investigation 1: Exponential Versus Linear Growth
Bernhard is looking into working for his mentor, Carl\(^1\), in a summer job in order to save for college.
Carl, being kind of a mathematical mind, decides to give Bernhard two options for payment to test his understanding of certain types of functions. He tells Bernhard he will either pay him \(\$50\) per day or pay him \(\$0.02\) if he works one day, \(\$0.04\) if he works two days, \(\$0.08\) if he works three days, etc. Carl will only pay Bernhard the total at the end of his time working!
Here are some sample values for the total amount of money Bernhard makes with each option.
Bernhard is looking into working for his mentor, Carl\(^1\), in a summer job in order to save for college.
Carl, being kind of a mathematical mind, decides to give Bernhard two options for payment to test his understanding of certain types of functions. He tells Bernhard he will either pay him \(\$50\) per day or pay him \(\$0.02\) if he works one day, \(\$0.04\) if he works two days, \(\$0.08\) if he works three days, etc. Carl will only pay Bernhard the total at the end of his time working!
Here are some sample values for the total amount of money Bernhard makes with each option.
Practice Questions
- If Bernhard can only work for \(14\) days, which option should he go with?
- Is there ever a situation in which Bernhard should pick Option 2?
- If Bernhard can only work for \(30\) days, which option should he go with?
- Write up an explanation for Bernhard to help him decide which option he should pick?
- Is there a situation in which Bernhard can pick either option and expect the same amount of money? (Plot two functions below that represent each option to assist you in answering this question.)
Bonus: Carl comes up with a third option to further challenge Bernhard’s analysis. In option 3, Carl pays, \(\$0.02\) at the end of day 1, \(\$0.04\) at the end of day 2, \(\$0.08\) at the end of day 3, etc. The difference between option 2 and 3 is that Bernhard gets paid every day in option 3, while in option 2 he just gets the payout at end of his time working. If option 3 is available, does this change your thoughts from earlier. To experiment with this try to add a third row to the table above to list these payouts, how much would Bernhard make after \(14\) days of work with option 3? (Hint: this bonus question can be revisited after the unit on sequences and series!)
Solutions to Practice Questions
Be careful when analyzing linear versus exponential situations!
\(1\)One of the most famous mathematicians of all time, Bernhard Riemann, was the doctoral student of another one of the most famous mathematicians of all time, Carl Friedrich Gauss.
Investigation 2: More background on Euler’s constant